@article{he2023deep,
  title={Deep neural networks and finite elements of any order on arbitrary dimensions},
  author={He, Juncai and Xu, Jinchao},
  journal={arXiv preprint arXiv:2312.14276},
  year={2023}
}
@article{he2018relu,
  title={ReLU deep neural networks and linear finite elements},
  author={He, Juncai and Li, Lin and Xu, Jinchao and Zheng, Chunyue},
  journal={arXiv preprint arXiv:1807.03973},
  year={2018}
}
@article{he2023optimal,
  title={On the optimal expressive power of relu dnns and its application in approximation with kolmogorov superposition theorem},
  author={He, Juncai},
  journal={arXiv preprint arXiv:2308.05509},
  year={2023}
}


@article{paszke2019pytorch,
  title={Pytorch: An imperative style, high-performance deep learning library},
  author={Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and others},
  journal={Advances in neural information processing systems},
  volume={32},
  year={2019}
}


@article{poluektov2023new,
  title={A new iterative method for construction of the Kolmogorov-Arnold representation},
  author={Poluektov, Michael and Polar, Andrew},
  journal={arXiv preprint arXiv:2305.08194},
  year={2023}
}


@article{ismayilova2024kolmogorov,
  title={On the Kolmogorov neural networks},
  author={Ismayilova, Aysu and Ismailov, Vugar E},
  journal={Neural Networks},
  pages={106333},
  year={2024},
  publisher={Elsevier}
}


@article{song2018optimizing,
  title={Optimizing kernel machines using deep learning},
  author={Song, Huan and Thiagarajan, Jayaraman J and Sattigeri, Prasanna and Spanias, Andreas},
  journal={IEEE transactions on neural networks and learning systems},
  volume={29},
  number={11},
  pages={5528--5540},
  year={2018},
  publisher={IEEE}
}



@article{wang2024multi,
  title={Multi-stage neural networks: Function approximator of machine precision},
  author={Wang, Yongji and Lai, Ching-Yao},
  journal={Journal of Computational Physics},
  pages={112865},
  year={2024},
  publisher={Elsevier}
}

@article{sun2021discerning,
  title={Discerning decision-making process of deep neural networks with hierarchical voting transformation},
  author={Sun, Ying and Zhu, Hengshu and Qin, Chuan and Zhuang, Fuzhen and He, Qing and Xiong, Hui},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={17221--17234},
  year={2021}
}


@article{zaheer2017deep,
  title={Deep sets},
  author={Zaheer, Manzil and Kottur, Satwik and Ravanbakhsh, Siamak and Poczos, Barnabas and Salakhutdinov, Russ R and Smola, Alexander J},
  journal={Advances in neural information processing systems},
  volume={30},
  year={2017}
}

@article{polar2021deep,
  title={A deep machine learning algorithm for construction of the Kolmogorov--Arnold representation},
  author={Polar, Andrew and Poluektov, Michael},
  journal={Engineering Applications of Artificial Intelligence},
  volume={99},
  pages={104137},
  year={2021},
  publisher={Elsevier}
}


@article{igelnik2003kolmogorov,
  title={Kolmogorov's spline network},
  author={Igelnik, Boris and Parikh, Neel},
  journal={IEEE transactions on neural networks},
  volume={14},
  number={4},
  pages={725--733},
  year={2003},
  publisher={IEEE}
}

@article{siegel2023optimal,
  title={Optimal approximation rates for deep ReLU neural networks on Sobolev and Besov spaces},
  author={Siegel, Jonathan W},
  journal={Journal of Machine Learning Research},
  volume={24},
  number={357},
  pages={1--52},
  year={2023}
}

@article{devore1989optimal,
  title={Optimal nonlinear approximation},
  author={DeVore, Ronald A and Howard, Ralph and Micchelli, Charles},
  journal={Manuscripta mathematica},
  volume={63},
  pages={469--478},
  year={1989},
  publisher={Springer}
}
@article{devore1993wavelet,
  title={Wavelet compression and nonlinear n-widths.},
  author={DeVore, Ronald A and Kyriazis, George and Leviatan, Dany and Tikhomirov, Vladimir M},
  journal={Adv. Comput. Math.},
  volume={1},
  number={2},
  pages={197--214},
  year={1993}
}
@article{yarotsky2017error,
  title={Error bounds for approximations with deep ReLU networks},
  author={Yarotsky, Dmitry},
  journal={Neural Networks},
  volume={94},
  pages={103--114},
  year={2017},
  publisher={Elsevier}
}
@article{siegel2024sharp,
  title={Sharp Lower Bounds on the Manifold Widths of Sobolev and Besov Spaces},
  author={Siegel, Jonathan W},
  journal={arXiv preprint arXiv:2402.04407},
  year={2024}
}
@article{bartlett2019nearly,
  title={Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks},
  author={Bartlett, Peter L and Harvey, Nick and Liaw, Christopher and Mehrabian, Abbas},
  journal={Journal of Machine Learning Research},
  volume={20},
  number={63},
  pages={1--17},
  year={2019}
}
@article{horowitz2007rate,
  title={Rate-optimal estimation for a general class of nonparametric regression models with unknown link functions},
  author={Horowitz, Joel L and Mammen, Enno},
  year={2007}
}

@article{schmidt2020nonparametric,
  title={Nonparametric regression using deep neural networks with ReLU activation function},
  author={Schmidt-Hieber, Johannes},
  year={2020}
}

@article{kohler2021rate,
  title={On the rate of convergence of fully connected deep neural network regression estimates},
  author={Kohler, Michael and Langer, Sophie},
  journal={The Annals of Statistics},
  volume={49},
  number={4},
  pages={2231--2249},
  year={2021},
  publisher={Institute of Mathematical Statistics}
}

@article{montanelli2020error,
  title={Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem},
  author={Montanelli, Hadrien and Yang, Haizhao},
  journal={Neural Networks},
  volume={129},
  pages={1--6},
  year={2020},
  publisher={Elsevier}
}

@article{lin2017does,
  title={Why does deep and cheap learning work so well?},
  author={Lin, Henry W and Tegmark, Max and Rolnick, David},
  journal={Journal of Statistical Physics},
  volume={168},
  pages={1223--1247},
  year={2017},
  publisher={Springer}
}

@misc{lu2024revisiting,
      title={Revisiting Neural Networks for Continual Learning: An Architectural Perspective}, 
      author={Aojun Lu and Tao Feng and Hangjie Yuan and Xiaotian Song and Yanan Sun},
      year={2024},
      eprint={2404.14829},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@article{chen2023exponentially,
  title={Exponentially convergent multiscale finite element method},
  author={Chen, Yifan and Hou, Thomas Y and Wang, Yixuan},
  journal={Communications on Applied Mathematics and Computation},
  pages={1--17},
  year={2023},
  publisher={Springer}
}
@article{ma2023unified,
  title={A unified framework for multiscale spectral generalized FEMs and low-rank approximations to multiscale PDEs},
  author={Ma, Chupeng},
  journal={arXiv preprint arXiv:2311.08761},
  year={2023}
}
@inproceedings{zhang2021multiscale,
  title={Multiscale invertible generative networks for high-dimensional Bayesian inference},
  author={Zhang, Shumao and Zhang, Pengchuan and Hou, Thomas Y},
  booktitle={International Conference on Machine Learning},
  pages={12632--12641},
  year={2021},
  organization={PMLR}
}
@article{xu2017algebraic,
  title={Algebraic multigrid methods},
  author={Xu, Jinchao and Zikatanov, Ludmil},
  journal={Acta Numerica},
  volume={26},
  pages={591--721},
  year={2017},
  publisher={Cambridge University Press}
}


@article{sitzmann2020implicit,
  title={Implicit neural representations with periodic activation functions},
  author={Sitzmann, Vincent and Martel, Julien and Bergman, Alexander and Lindell, David and Wetzstein, Gordon},
  journal={Advances in neural information processing systems},
  volume={33},
  pages={7462--7473},
  year={2020}
}


@incollection{leni2013kolmogorov,
  title={The kolmogorov spline network for image processing},
  author={Leni, Pierre-Emmanuel and Fougerolle, Yohan D and Truchetet, Fr{\'e}d{\'e}ric},
  booktitle={Image Processing: Concepts, Methodologies, Tools, and Applications},
  pages={54--78},
  year={2013},
  publisher={IGI Global}
}


@article{lai2021kolmogorov,
  title={The kolmogorov superposition theorem can break the curse of dimensionality when approximating high dimensional functions},
  author={Lai, Ming-Jun and Shen, Zhaiming},
  journal={arXiv preprint arXiv:2112.09963},
  year={2021}
}


@article{lin1993realization,
  title={On the realization of a Kolmogorov network},
  author={Lin, Ji-Nan and Unbehauen, Rolf},
  journal={Neural Computation},
  volume={5},
  number={1},
  pages={18--20},
  year={1993},
  publisher={MIT Press}
}


@inproceedings{koppen2002training,
  title={On the training of a Kolmogorov Network},
  author={K{\"o}ppen, Mario},
  booktitle={Artificial Neural Networks—ICANN 2002: International Conference Madrid, Spain, August 28--30, 2002 Proceedings 12},
  pages={474--479},
  year={2002},
  organization={Springer}
}

@article{sprecher2002space,
  title={Space-filling curves and Kolmogorov superposition-based neural networks},
  author={Sprecher, David A and Draghici, Sorin},
  journal={Neural Networks},
  volume={15},
  number={1},
  pages={57--67},
  year={2002},
  publisher={Elsevier}
}

@article{kolmogorov,
  author  = "A.N. Kolmogorov",
  title   = "On the Representation of continuous functions of several variables as superpositions of continuous functions of a smaller number of variables.",
  journal = "Dokl. Akad. Nauk",
  year    = 1956,
  volume  = "108",
  number  = "2",
}


@article{braun2009constructive,
  title={On a constructive proof of Kolmogorov’s superposition theorem},
  author={Braun, J{\"u}rgen and Griebel, Michael},
  journal={Constructive approximation},
  volume={30},
  pages={653--675},
  year={2009},
  publisher={Springer}
}


@article{vaswani2017attention,
  title={Attention is all you need},
  author={Vaswani, Ashish and Shazeer, Noam and Parmar, Niki and Uszkoreit, Jakob and Jones, Llion and Gomez, Aidan N and Kaiser, {\L}ukasz and Polosukhin, Illia},
  journal={Advances in neural information processing systems},
  volume={30},
  year={2017}
}


@article{hornik1989multilayer,
  title={Multilayer feedforward networks are universal approximators},
  author={Hornik, Kurt and Stinchcombe, Maxwell and White, Halbert},
  journal={Neural networks},
  volume={2},
  number={5},
  pages={359--366},
  year={1989},
  publisher={Elsevier}
}


@article{cybenko1989approximation,
  title={Approximation by superpositions of a sigmoidal function},
  author={Cybenko, George},
  journal={Mathematics of control, signals and systems},
  volume={2},
  number={4},
  pages={303--314},
  year={1989},
  publisher={Springer}
}


@book{haykin1994neural,
  title={Neural networks: a comprehensive foundation},
  author={Haykin, Simon},
  year={1994},
  publisher={Prentice Hall PTR}
}


@inproceedings{
mundhenk2021symbolic,
title={Symbolic Regression via Deep Reinforcement Learning Enhanced Genetic Programming Seeding},
author={Terrell N. Mundhenk and Mikel Landajuela and Ruben Glatt and Claudio P. Santiago and Daniel faissol and Brenden K. Petersen},
booktitle={Advances in Neural Information Processing Systems},
editor={A. Beygelzimer and Y. Dauphin and P. Liang and J. Wortman Vaughan},
year={2021},
url={https://openreview.net/forum?id=tjwQaOI9tdy}
}


@article{udrescu2020ai2,
  title={AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity},
  author={Udrescu, Silviu-Marian and Tan, Andrew and Feng, Jiahai and Neto, Orisvaldo and Wu, Tailin and Tegmark, Max},
  journal={Advances in Neural Information Processing Systems},
  volume={33},
  pages={4860--4871},
  year={2020}
}


@article{udrescu2020ai,
  title={AI Feynman: A physics-inspired method for symbolic regression},
  author={Udrescu, Silviu-Marian and Tegmark, Max},
  journal={Science Advances},
  volume={6},
  number={16},
  pages={eaay2631},
  year={2020},
  publisher={American Association for the Advancement of Science}
}


@article{dugan2020occamnet,
  title={OccamNet: A Fast Neural Model for Symbolic Regression at Scale},
  author={Dugan, Owen and Dangovski, Rumen and Costa, Allan and Kim, Samuel and Goyal, Pawan and Jacobson, Joseph and Solja{\v{c}}i{\'c}, Marin},
  journal={arXiv preprint arXiv:2007.10784},
  year={2020}
}


@article{martius2016extrapolation,
  title={Extrapolation and learning equations},
  author={Martius, Georg and Lampert, Christoph H},
  journal={arXiv preprint arXiv:1610.02995},
  year={2016}
}


@article{cranmer2023interpretable,
  title={Interpretable machine learning for science with PySR and SymbolicRegression. jl},
  author={Cranmer, Miles},
  journal={arXiv preprint arXiv:2305.01582},
  year={2023}
}


@misc{gplearn,
  title = {GPLearn},
  howpublished = {\url{https://github.com/trevorstephens/gplearn}},
  note = {Accessed: 2024-04-19}
}

@article{Dubckov2011EureqaSR,
  title={Eureqa: software review},
  author={Ren{\'a}ta Dubc{\'a}kov{\'a}},
  journal={Genetic Programming and Evolvable Machines},
  year={2011},
  volume={12},
  pages={173-178},
  url={https://api.semanticscholar.org/CorpusID:36698573}
}

@article{meunier2010modular,
  title={Modular and hierarchically modular organization of brain networks},
  author={Meunier, David and Lambiotte, Renaud and Bullmore, Edward T},
  journal={Frontiers in neuroscience},
  volume={4},
  pages={7572},
  year={2010},
  publisher={Frontiers}
}


@article{kolb1998brain,
  title={Brain plasticity and behavior},
  author={Kolb, Bryan and Whishaw, Ian Q},
  journal={Annual review of psychology},
  volume={49},
  number={1},
  pages={43--64},
  year={1998},
  publisher={Annual Reviews 4139 El Camino Way, PO Box 10139, Palo Alto, CA 94303-0139, USA}
}


@inproceedings{kemker2018measuring,
  title={Measuring catastrophic forgetting in neural networks},
  author={Kemker, Ronald and McClure, Marc and Abitino, Angelina and Hayes, Tyler and Kanan, Christopher},
  booktitle={Proceedings of the AAAI conference on artificial intelligence},
  volume={32},
  year={2018}
}


@article{kirkpatrick2017overcoming,
  title={Overcoming catastrophic forgetting in neural networks},
  author={Kirkpatrick, James and Pascanu, Razvan and Rabinowitz, Neil and Veness, Joel and Desjardins, Guillaume and Rusu, Andrei A and Milan, Kieran and Quan, John and Ramalho, Tiago and Grabska-Barwinska, Agnieszka and others},
  journal={Proceedings of the national academy of sciences},
  volume={114},
  number={13},
  pages={3521--3526},
  year={2017},
  publisher={National Acad Sciences}
}


@article{cunningham2023sparse,
  title={Sparse autoencoders find highly interpretable features in language models},
  author={Cunningham, Hoagy and Ewart, Aidan and Riggs, Logan and Huben, Robert and Sharkey, Lee},
  journal={arXiv preprint arXiv:2309.08600},
  year={2023}
}


@article{elhage2022solu,
   title={Softmax Linear Units},
   author={Elhage, Nelson and Hume, Tristan and Olsson, Catherine and Nanda, Neel and Henighan, Tom and Johnston, Scott and ElShowk, Sheer and Joseph, Nicholas and DasSarma, Nova and Mann, Ben and Hernandez, Danny and Askell, Amanda and Ndousse, Kamal and Jones, Andy and Drain, Dawn and Chen, Anna and Bai, Yuntao and Ganguli, Deep and Lovitt, Liane and Hatfield-Dodds, Zac and Kernion, Jackson and Conerly, Tom and Kravec, Shauna and Fort, Stanislav and Kadavath, Saurav and Jacobson, Josh and Tran-Johnson, Eli and Kaplan, Jared and Clark, Jack and Brown, Tom and McCandlish, Sam and Amodei, Dario and Olah, Christopher},
   year={2022},
   journal={Transformer Circuits Thread},
   note={https://transformer-circuits.pub/2022/solu/index.html}
}


@article{liu2023seeing,
  title={Seeing is believing: Brain-inspired modular training for mechanistic interpretability},
  author={Liu, Ziming and Gan, Eric and Tegmark, Max},
  journal={Entropy},
  volume={26},
  number={1},
  pages={41},
  year={2023},
  publisher={MDPI}
}


@inproceedings{
zhong2023the,
title={The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks},
author={Ziqian Zhong and Ziming Liu and Max Tegmark and Jacob Andreas},
booktitle={Thirty-seventh Conference on Neural Information Processing Systems},
year={2023},
url={https://openreview.net/forum?id=S5wmbQc1We}
}


@inproceedings{
nanda2023progress,
title={Progress measures for grokking via mechanistic interpretability},
author={Neel Nanda and Lawrence Chan and Tom Lieberum and Jess Smith and Jacob Steinhardt},
booktitle={The Eleventh International Conference on Learning Representations },
year={2023},
url={https://openreview.net/forum?id=9XFSbDPmdW}
}


@article{elhage2022toy,
  title={Toy models of superposition},
  author={Elhage, Nelson and Hume, Tristan and Olsson, Catherine and Schiefer, Nicholas and Henighan, Tom and Kravec, Shauna and Hatfield-Dodds, Zac and Lasenby, Robert and Drain, Dawn and Chen, Carol and others},
  journal={arXiv preprint arXiv:2209.10652},
  year={2022}
}


@inproceedings{
wang2023interpretability,
title={Interpretability in the Wild: a Circuit for Indirect Object Identification in {GPT}-2 Small},
author={Kevin Ro Wang and Alexandre Variengien and Arthur Conmy and Buck Shlegeris and Jacob Steinhardt},
booktitle={The Eleventh International Conference on Learning Representations },
year={2023},
url={https://openreview.net/forum?id=NpsVSN6o4ul}
}



@article{meng2022locating,
  title={Locating and editing factual associations in GPT},
  author={Meng, Kevin and Bau, David and Andonian, Alex and Belinkov, Yonatan},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={17359--17372},
  year={2022}
}



@article{olsson2022context,
  title={In-context learning and induction heads},
  author={Olsson, Catherine and Elhage, Nelson and Nanda, Neel and Joseph, Nicholas and DasSarma, Nova and Henighan, Tom and Mann, Ben and Askell, Amanda and Bai, Yuntao and Chen, Anna and others},
  journal={arXiv preprint arXiv:2209.11895},
  year={2022}
}


@article{song2024resource,
  title={A Resource Model For Neural Scaling Law},
  author={Song, Jinyeop and Liu, Ziming and Tegmark, Max and Gore, Jeff},
  journal={arXiv preprint arXiv:2402.05164},
  year={2024}
}


@inproceedings{
michaud2023the,
title={The Quantization Model of Neural Scaling},
author={Eric J Michaud and Ziming Liu and Uzay Girit and Max Tegmark},
booktitle={Thirty-seventh Conference on Neural Information Processing Systems},
year={2023},
url={https://openreview.net/forum?id=3tbTw2ga8K}
}


@article{bahri2021explaining,
  title={Explaining neural scaling laws},
  author={Bahri, Yasaman and Dyer, Ethan and Kaplan, Jared and Lee, Jaehoon and Sharma, Utkarsh},
  journal={arXiv preprint arXiv:2102.06701},
  year={2021}
}


@article{hestness2017deep,
  title={Deep learning scaling is predictable, empirically},
  author={Hestness, Joel and Narang, Sharan and Ardalani, Newsha and Diamos, Gregory and Jun, Heewoo and Kianinejad, Hassan and Patwary, Md Mostofa Ali and Yang, Yang and Zhou, Yanqi},
  journal={arXiv preprint arXiv:1712.00409},
  year={2017}
}

@inproceedings{
gordon2021data,
title={Data and Parameter Scaling Laws for Neural Machine Translation},
author={Mitchell A Gordon and Kevin Duh and Jared Kaplan},
booktitle={ACL Rolling Review - May 2021},
year={2021},
url={https://openreview.net/forum?id=IKA7MLxsLSu}
}


@article{henighan2020scaling,
  title={Scaling laws for autoregressive generative modeling},
  author={Henighan, Tom and Kaplan, Jared and Katz, Mor and Chen, Mark and Hesse, Christopher and Jackson, Jacob and Jun, Heewoo and Brown, Tom B and Dhariwal, Prafulla and Gray, Scott and others},
  journal={arXiv preprint arXiv:2010.14701},
  year={2020}
}

@article{kaplan2020scaling,
  title={Scaling laws for neural language models},
  author={Kaplan, Jared and McCandlish, Sam and Henighan, Tom and Brown, Tom B and Chess, Benjamin and Child, Rewon and Gray, Scott and Radford, Alec and Wu, Jeffrey and Amodei, Dario},
  journal={arXiv preprint arXiv:2001.08361},
  year={2020}
}

@book{petersen2006riemannian,
  title={Riemannian Geometry},
  author={Petersen, P.},
  isbn={9780387294032},
  lccn={97005786},
  series={Graduate Texts in Mathematics},
  url={https://books.google.com/books?id=9cekXdo52hEC},
  year={2006},
  publisher={Springer New York}
}

@article{Gukov:2020qaj,
    author = "Gukov, Sergei and Halverson, James and Ruehle, Fabian and Su\l{}kowski, Piotr",
    title = "{Learning to Unknot}",
    eprint = "2010.16263",
    archivePrefix = "arXiv",
    primaryClass = "math.GT",
    reportNumber = "CALT-2020-046. CERN-TH-2020-179",
    doi = "10.1088/2632-2153/abe91f",
    journal = "Mach. Learn. Sci. Tech.",
    volume = "2",
    number = "2",
    pages = "025035",
    year = "2021"
}

@misc{kauffman2020rectangular,
      title={Rectangular knot diagrams classification with deep learning}, 
      author={L. H. Kauffman and N. E. Russkikh and I. A. Taimanov},
      year={2020},
      eprint={2011.03498},
      archivePrefix={arXiv},
      primaryClass={math.GT}
}

@misc{gukov2023searching,
      title={Searching for ribbons with machine learning}, 
      author={Sergei Gukov and James Halverson and Ciprian Manolescu and Fabian Ruehle},
      year={2023},
      eprint={2304.09304},
      archivePrefix={arXiv},
      primaryClass={math.GT}
}

@article{hughes2020neural,
  title={A neural network approach to predicting and computing knot invariants},
  author={Hughes, Mark C},
  journal={Journal of Knot Theory and Its Ramifications},
  volume={29},
  number={03},
  pages={2050005},
  year={2020},
  publisher={World Scientific}
}

@article{Craven:2020bdz,
    author = "Craven, Jessica and Jejjala, Vishnu and Kar, Arjun",
    title = "{Disentangling a deep learned volume formula}",
    eprint = "2012.03955",
    archivePrefix = "arXiv",
    primaryClass = "hep-th",
    doi = "10.1007/JHEP06(2021)040",
    journal = "JHEP",
    volume = "06",
    pages = "040",
    year = "2021"
}

@article{Craven:2022cxe,
    author = "Craven, Jessica and Hughes, Mark and Jejjala, Vishnu and Kar, Arjun",
    title = "{Illuminating new and known relations between knot invariants}",
    eprint = "2211.01404",
    archivePrefix = "arXiv",
    primaryClass = "math.GT",
    month = "11",
    year = "2022"
}

@article{Ruehle:2020jrk,
    author = "Ruehle, Fabian",
    title = "{Data science applications to string theory}",
    doi = "10.1016/j.physrep.2019.09.005",
    journal = "Phys. Rept.",
    volume = "839",
    pages = "1--117",
    year = "2020"
}

@book{he2023machine,
  title={Machine Learning in Pure Mathematics and Theoretical Physics},
  author={He, Y.H.},
  isbn={9781800613690},
  lccn={2022058911},
  series={G - Reference,Information and Interdisciplinary Subjects Series},
  url={https://books.google.com/books?id=6a5gzwEACAAJ},
  year={2023},
  publisher={World Scientific}
}


@article{Gukov:2024aaa,
	author = {Gukov, Sergei and Halverson, James and Ruehle, Fabian},
	doi = {10.1038/s42254-024-00709-0},
	id = {Gukov2024},
	isbn = {2522-5820},
	journal = {Nature Reviews Physics},
	title = {Rigor with machine learning from field theory to the Poincar{\'e}conjecture},
	url = {https://doi.org/10.1038/s42254-024-00709-0},
	year = {2024},
	bdsk-url-1 = {https://doi.org/10.1038/s42254-024-00709-0}
}


@article{davies2021advancing,
  title={Advancing mathematics by guiding human intuition with AI},
  author={Davies, Alex and Veli{\v{c}}kovi{\'c}, Petar and Buesing, Lars and Blackwell, Sam and Zheng, Daniel and Toma{\v{s}}ev, Nenad and Tanburn, Richard and Battaglia, Peter and Blundell, Charles and Juh{\'a}sz, Andr{\'a}s and others},
  journal={Nature},
  volume={600},
  number={7887},
  pages={70--74},
  year={2021},
  publisher={Nature Publishing Group}
}

@article{poggio2020theoretical,
  title={Theoretical issues in deep networks},
  author={Poggio, Tomaso and Banburski, Andrzej and Liao, Qianli},
  journal={Proceedings of the National Academy of Sciences},
  volume={117},
  number={48},
  pages={30039--30045},
  year={2020},
  publisher={National Acad Sciences}
}

@article{michaud2023precision,
  title={Precision machine learning},
  author={Michaud, Eric J and Liu, Ziming and Tegmark, Max},
  journal={Entropy},
  volume={25},
  number={1},
  pages={175},
  year={2023},
  publisher={MDPI}
}


@article{sharma2020neural,
  title={A neural scaling law from the dimension of the data manifold},
  author={Sharma, Utkarsh and Kaplan, Jared},
  journal={arXiv preprint arXiv:2004.10802},
  year={2020}
}


@article{xu2015nonlinear,
  title={Nonlinear material design using principal stretches},
  author={Xu, Hongyi and Sin, Funshing and Zhu, Yufeng and Barbi{\v{c}}, Jernej},
  journal={ACM Transactions on Graphics (TOG)},
  volume={34},
  number={4},
  pages={1--11},
  year={2015},
  publisher={ACM New York, NY, USA}
}


@inproceedings{aziznejad2019deep,
  title={Deep spline networks with control of Lipschitz regularity},
  author={Aziznejad, Shayan and Unser, Michael},
  booktitle={ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)},
  pages={3242--3246},
  year={2019},
  organization={IEEE}
}


@article{bohra2020learning,
  title={Learning activation functions in deep (spline) neural networks},
  author={Bohra, Pakshal and Campos, Joaquim and Gupta, Harshit and Aziznejad, Shayan and Unser, Michael},
  journal={IEEE Open Journal of Signal Processing},
  volume={1},
  pages={295--309},
  year={2020},
  publisher={IEEE}
}
@article{cho2024separable,
  title={Separable physics-informed neural networks},
  author={Cho, Junwoo and Nam, Seungtae and Yang, Hyunmo and Yun, Seok-Bae and Hong, Youngjoon and Park, Eunbyung},
  journal={Advances in Neural Information Processing Systems},
  volume={36},
  year={2024}
}
@article{zhang2022neural,
  title={Neural network architecture beyond width and depth},
  author={Zhang, Shijun and Shen, Zuowei and Yang, Haizhao},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={5669--5681},
  year={2022}
}


@article{fakhoury2022exsplinet,
  title={ExSpliNet: An interpretable and expressive spline-based neural network},
  author={Fakhoury, Daniele and Fakhoury, Emanuele and Speleers, Hendrik},
  journal={Neural Networks},
  volume={152},
  pages={332--346},
  year={2022},
  publisher={Elsevier}
}


@article{goyal2019learning,
  title={Learning activation functions: A new paradigm for understanding neural networks},
  author={Goyal, Mohit and Goyal, Rajan and Lall, Brejesh},
  journal={arXiv preprint arXiv:1906.09529},
  year={2019}
}

@article{bingham2022discovering,
  title={Discovering parametric activation functions},
  author={Bingham, Garrett and Miikkulainen, Risto},
  journal={Neural Networks},
  volume={148},
  pages={48--65},
  year={2022},
  publisher={Elsevier}
}


@article{ramachandran2017searching,
  title={Searching for activation functions},
  author={Ramachandran, Prajit and Zoph, Barret and Le, Quoc V},
  journal={arXiv preprint arXiv:1710.05941},
  year={2017}
}


@article{agarwal2021neural,
  title={Neural additive models: Interpretable machine learning with neural nets},
  author={Agarwal, Rishabh and Melnick, Levi and Frosst, Nicholas and Zhang, Xuezhou and Lengerich, Ben and Caruana, Rich and Hinton, Geoffrey E},
  journal={Advances in neural information processing systems},
  volume={34},
  pages={4699--4711},
  year={2021}
}

@article{schmidt2021kolmogorov,
  title={The Kolmogorov--Arnold representation theorem revisited},
  author={Schmidt-Hieber, Johannes},
  journal={Neural networks},
  volume={137},
  pages={119--126},
  year={2021},
  publisher={Elsevier}
}


@article{poggio2022deep,
  title={How deep sparse networks avoid the curse of dimensionality: Efficiently computable functions are compositionally sparse},
  author={Poggio, Tomaso},
  journal={CBMM Memo},
  volume={10},
  pages={2022},
  year={2022}
}
@inproceedings{kolmogorov1957representation,
  title={On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition},
  author={Kolmogorov, Andrei Nikolaevich},
  booktitle={Doklady Akademii Nauk},
  volume={114},
  pages={953--956},
  year={1957},
  organization={Russian Academy of Sciences}
}
@book{de1978practical,
  title={A practical guide to splines},
  author={De Boor, Carl},
  volume={27},
  year={1978},
  publisher={springer-verlag New York}
}

@article{lu2021learning,
  title={Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators},
  author={Lu, Lu and Jin, Pengzhan and Pang, Guofei and Zhang, Zhongqiang and Karniadakis, George Em},
  journal={Nature machine intelligence},
  volume={3},
  number={3},
  pages={218--229},
  year={2021},
  publisher={Nature Publishing Group UK London}
}

@article{raissi2019physics,
  title={Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},
  author={Raissi, Maziar and Perdikaris, Paris and Karniadakis, George E},
  journal={Journal of Computational physics},
  volume={378},
  pages={686--707},
  year={2019},
  publisher={Elsevier}
}

@article{karniadakis2021physics,
  title={Physics-informed machine learning},
  author={Karniadakis, George Em and Kevrekidis, Ioannis G and Lu, Lu and Perdikaris, Paris and Wang, Sifan and Yang, Liu},
  journal={Nature Reviews Physics},
  volume={3},
  number={6},
  pages={422--440},
  year={2021},
  publisher={Nature Publishing Group}
}

@article{yu2018deep,
  title={The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems},
  author={Yu, Bing and others},
  journal={Communications in Mathematics and Statistics},
  volume={6},
  number={1},
  pages={1--12},
  year={2018},
  publisher={Springer}
}

@article{li2021physics,
  title={Physics-informed neural operator for learning partial differential equations},
  author={Li, Zongyi and Zheng, Hongkai and Kovachki, Nikola and Jin, David and Chen, Haoxuan and Liu, Burigede and Azizzadenesheli, Kamyar and Anandkumar, Anima},
  journal={ACM/JMS Journal of Data Science},
  year={2021},
  publisher={ACM New York, NY}
}

@article{kovachki2023neural,
  title={Neural operator: Learning maps between function spaces with applications to pdes},
  author={Kovachki, Nikola and Li, Zongyi and Liu, Burigede and Azizzadenesheli, Kamyar and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima},
  journal={Journal of Machine Learning Research},
  volume={24},
  number={89},
  pages={1--97},
  year={2023}
}

@article{li2020fourier,
  title={Fourier neural operator for parametric partial differential equations},
  author={Li, Zongyi and Kovachki, Nikola and Azizzadenesheli, Kamyar and Liu, Burigede and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima},
  journal={arXiv preprint arXiv:2010.08895},
  year={2020}
}





@article{maust2022fourier,
  title={Fourier continuation for exact derivative computation in physics-informed neural operators},
  author={Maust, Haydn and Li, Zongyi and Wang, Yixuan and Leibovici, Daniel and Bruno, Oscar and Hou, Thomas and Anandkumar, Anima},
  journal={arXiv preprint arXiv:2211.15960},
  year={2022}
}

@article{ME_biddle2010predicted,
  title={Predicted mobility edges in one-dimensional incommensurate optical lattices: An exactly solvable model of Anderson localization},
  author={Biddle, J and Sarma, S Das},
  journal={Physical review letters},
  volume={104},
  number={7},
  pages={070601},
  year={2010},
  publisher={APS}
}

@article{ME_ganeshan2015nearest,
  title={Nearest neighbor tight binding models with an exact mobility edge in one dimension},
  author={Ganeshan, Sriram and Pixley, JH and Sarma, S Das},
  journal={Physical review letters},
  volume={114},
  number={14},
  pages={146601},
  year={2015},
  publisher={APS}
}

@article{ME_wang2020one,
  title={One-dimensional quasiperiodic mosaic lattice with exact mobility edges},
  author={Wang, Yucheng and Xia, Xu and Zhang, Long and Yao, Hepeng and Chen, Shu and You, Jiangong and Zhou, Qi and Liu, Xiong-Jun},
  journal={Physical Review Letters},
  volume={125},
  number={19},
  pages={196604},
  year={2020},
  publisher={APS}
}

@article{ME_an2021interactions,
  title={Interactions and mobility edges: Observing the generalized aubry-andr{\'e} model},
  author={An, Fangzhao Alex and Padavi{\'c}, Karmela and Meier, Eric J and Hegde, Suraj and Ganeshan, Sriram and Pixley, JH and Vishveshwara, Smitha and Gadway, Bryce},
  journal={Physical review letters},
  volume={126},
  number={4},
  pages={040603},
  year={2021},
  publisher={APS}
}

@article{ME_duthie2021self,
  title={Self-consistent theory of mobility edges in quasiperiodic chains},
  author={Duthie, Alexander and Roy, Sthitadhi and Logan, David E},
  journal={Physical Review B},
  volume={103},
  number={6},
  pages={L060201},
  year={2021},
  publisher={APS}
}

@article{ME_wang2021duality,
  title={Duality between two generalized Aubry-Andr{\'e} models with exact mobility edges},
  author={Wang, Yucheng and Xia, Xu and Wang, Yongjian and Zheng, Zuohuan and Liu, Xiong-Jun},
  journal={Physical Review B},
  volume={103},
  number={17},
  pages={174205},
  year={2021},
  publisher={APS}
}

@article{ME_zhou2023exact,
  title={Exact new mobility edges between critical and localized states},
  author={Zhou, Xin-Chi and Wang, Yongjian and Poon, Ting-Fung Jeffrey and Zhou, Qi and Liu, Xiong-Jun},
  journal={Physical Review Letters},
  volume={131},
  number={17},
  pages={176401},
  year={2023},
  publisher={APS}
}

@article{vaidya2023reentrant,
  title={Reentrant delocalization transition in one-dimensional photonic quasicrystals},
  author={Vaidya, Sachin and J{\"o}rg, Christina and Linn, Kyle and Goh, Megan and Rechtsman, Mikael C},
  journal={Physical Review Research},
  volume={5},
  number={3},
  pages={033170},
  year={2023},
  publisher={APS}
}

@article{anderson1958absence,
  title={Absence of diffusion in certain random lattices},
  author={Anderson, Philip W},
  journal={Physical review},
  volume={109},
  number={5},
  pages={1492},
  year={1958},
  publisher={APS}
}

@article{thouless1972relation,
  title={A relation between the density of states and range of localization for one dimensional random systems},
  author={Thouless, David J},
  journal={Journal of Physics C: Solid State Physics},
  volume={5},
  number={1},
  pages={77},
  year={1972},
  publisher={IOP Publishing}
}

@article{abrahams1979scaling,
  title={Scaling theory of localization: Absence of quantum diffusion in two dimensions},
  author={Abrahams, Elihu and Anderson, PW and Licciardello, DC and Ramakrishnan, TV},
  journal={Physical Review Letters},
  volume={42},
  number={10},
  pages={673},
  year={1979},
  publisher={APS}
}

@article{segev2013anderson,
  title={Anderson localization of light},
  author={Segev, Mordechai and Silberberg, Yaron and Christodoulides, Demetrios N},
  journal={Nature Photonics},
  volume={7},
  number={3},
  pages={197--204},
  year={2013},
  publisher={Nature Publishing Group UK London}
}

@article{john1987strong,
  title={Strong localization of photons in certain disordered dielectric superlattices},
  author={John, Sajeev},
  journal={Physical review letters},
  volume={58},
  number={23},
  pages={2486},
  year={1987},
  publisher={APS}
}

@article{lahini2009observation,
  title={Observation of a localization transition in quasiperiodic photonic lattices},
  author={Lahini, Yoav and Pugatch, Rami and Pozzi, Francesca and Sorel, Marc and Morandotti, Roberto and Davidson, Nir and Silberberg, Yaron},
  journal={Physical review letters},
  volume={103},
  number={1},
  pages={013901},
  year={2009},
  publisher={APS}
}

@article{vardeny2013optics,
  title={Optics of photonic quasicrystals},
  author={Vardeny, Z Valy and Nahata, Ajay and Agrawal, Amit},
  journal={Nature photonics},
  volume={7},
  number={3},
  pages={177--187},
  year={2013},
  publisher={Nature Publishing Group UK London}
}

@article{de2016absence,
  title={Absence of many-body mobility edges},
  author={De Roeck, Wojciech and Huveneers, Francois and M{\"u}ller, Markus and Schiulaz, Mauro},
  journal={Physical Review B},
  volume={93},
  number={1},
  pages={014203},
  year={2016},
  publisher={APS}
}

@article{li2015many,
  title={Many-body localization and quantum nonergodicity in a model with a single-particle mobility edge},
  author={Li, Xiaopeng and Ganeshan, Sriram and Pixley, JH and Sarma, S Das},
  journal={Physical review letters},
  volume={115},
  number={18},
  pages={186601},
  year={2015},
  publisher={APS}
}

@article{lagendijk2009fifty,
  title={Fifty years of Anderson localization},
  author={Lagendijk, Ad and Tiggelen, Bart van and Wiersma, Diederik S},
  journal={Physics today},
  volume={62},
  number={8},
  pages={24--29},
  year={2009},
  publisher={AIP Publishing}
}